exponential_spacetime {GpGp}

Spatial-Temporal exponential covariance function

Description

From a matrix of locations and covariance parameters of the form (variance, range_1, range_2, nugget), return the square matrix of all pairwise covariances.

Usage

exponential_spacetime(covparms, locs)

d_exponential_spacetime(covparms, locs)

Arguments

covparms

A vector with covariance parameters in the form (variance, range_1, range_2, nugget). range_1 is the spatial range, and range_2 is the temporal range.

locs

A matrix with n rows and d+1 columns. Each row of locs is a point in R^(d+1). The first d columns should contain the spatial coordinates. The last column contains the times.

Value

A matrix with n rows and n columns, with the i,j entry containing the covariance between observations at locs[i,] and locs[j,].

Functions

• d_exponential_spacetime(): Derivatives with respect to parameters

Parameterization

The covariance parameter vector is (variance, range_1, range_2, nugget). The covariance function is parameterized as

M(x,y) = \sigma^2 exp( - || D^{-1}(x - y) || )

where D is a diagonal matrix with (range_1, ..., range_1, range_2) on the diagonals. The nugget value \sigma^2 \tau^2 is added to the diagonal of the covariance matrix. NOTE: the nugget is \sigma^2 \tau^2 , not \tau^2 .